nLab
bubble of nothing

Contents

Context

Gravity

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Quantum field theory

String theory

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

A bubble of nothing in the sense of Witten 82 is an instability of the Kaluza-Klein vacuum where a gravitational instanton mediates the decay of a 55-dimensional Euclidean Schwarzschild spacetime into nothing: the hole in space grows at the speed of light until it hits null infinity?, leading to the “annihilation” of spacetime.

Construction of the solution

Let us start from the spacetime ( 4B R 3)×S 1(\mathbb{R}^{4}-B_R^3)\times S^1, where B R 3B_R^3 is a ball of radius RR and S 1S^1 is a circle of radius R 5R_5, equipped with the following Euclidean Schwarzschild metric:

g=r 2vol S 3+dr 21R 2r 2+R 5 2(1R 2r 2)dθ 2, g \;=\; r^2\mathrm{vol}_{S^3} + \frac{\mathrm{d}r^2}{1-\frac{R^2}{r^2}} + R_5^2 \left(1-\frac{R^2}{r^2}\right)\mathrm{d}\theta^2,

where θ\theta is the coordinate on the circle and rr is the radius of 4\mathbb{R}^{4}.

Let the volume of the 33-sphere be vol S 3=dχ 2+sin(χ)vol S 2\mathrm{vol}_{S^3} = \mathrm{d}\chi^2 + \sin(\chi)\mathrm{vol}_{S^2}. We can now go back to the Minkowski signature by the Wick rotation ψ:=iχ+iπ2\psi := -i\chi + i\frac{\pi}{2}. Thus, we have

g=r 2dψ 2+dr 21R 2r 2+r 2cosh 2(ψ)vol S 2+R 5 2(1R 2r 2)dθ 2. g \;=\; -r^2\mathrm{d}\psi^2 + \frac{\mathrm{d}r^2}{1-\frac{R^2}{r^2}} + r^2\cosh^2(\psi)\mathrm{vol}_{S^2} + R_5^2 \left(1-\frac{R^2}{r^2}\right)\mathrm{d}\theta^2.

Properties

By change of coordinates ρ:=rcosh(ψ)\rho := r\cosh(\psi) and t:=rsinh(ψ)t:=r\sinh(\psi), we see that at large radius the metric goes to the Minkowski metric

lim rg=dt 2+dρ 2+ρ 2vol S 2+R 5 2dθ 2.\lim_{r\rightarrow \infty}g \;=\; -\mathrm{d}t^2 +\mathrm{d}\rho^2 + \rho^2\mathrm{vol}_{S^2}+R_5^2\mathrm{d}\theta^2.

However, the coordinates (t,ρ)(t,\rho) parametrize all the 44-dimensional base manifold but the region {ρ 2t 2R}\{ \rho^2 - t^2 \leq R \} of the bubble. The boundary of the bubble in the 44-dimensional base space has a radius which grows with time as

ρ BON(t)=R 2+t 2. \rho_{\mathrm{BON}}(t) \;=\;\sqrt{R^2+t^2}.

Moreover, the effective size of the circle compactification will be depend on rr by

R 5(r)=R 51R 2r 2, R_{5}(r) \;=\; R_5\sqrt{1-\frac{R^2}{r^2}},

so that it goes to 00 for rRr\rightarrow R and it goes to R 5R_5 for for rr\rightarrow \infty.

References

Last revised on April 21, 2021 at 07:48:41. See the history of this page for a list of all contributions to it.